Volume 00, Number 0, Pages 000000 S 0002-9939(XX)0000-0
ON THE SELF-SIMILARITY PROBLEM FOR GAUSSIAN-KRONECKER FLOWS
KRZYSZTOF FRCZEK, JOANNA KUAGA, AND MARIUSZ LEMACZYK (Communicated by Nimish Shah)
Abstract. It is shown that a countable symmetric multiplicative subgroup G = −H ∪ H with H ⊂ R∗+ is the group of self-similarities of a
Gaussian-Kronecker ow if and only if H is additively Q-independent. In particular, a real number s 6= ±1 is a scale of self-similarity of a Gaussian-Kronecker ow if and only if s is transcendental. We also show that each countable symmetric subgroup of R∗ can be realized as the group of self-similarities of a simple
spectrum Gaussian ow having the Foia³-Stratila property.
1. Introduction
Assume that T = (Tt)t∈R is a (measurable) measure-preserving ow acting on
a standard probability Borel space (X, B, µ). Given s ∈ R∗, one says that it is
a scale of self-similarity of T if T is isomorphic to Ts := (Tst)t∈R. Denote by
I(T ) the set of all scales of self-similarities of T . Then T is called self-similar if I(T ) 6= {±1}. Classical examples of self-similar ows are given by horocycle ows where I(T ) equals either R∗ or R∗
+ [19]. A systematic study of the problem
of self-similarity has been done recently in [4] and [6]. In particular, I(T ) turns out to be a multiplicative subgroup of R∗ ([6]) which is Borel ([4]), and one of
the main problems in this domain is to classify all Borel subgroups of R∗ that
may appear as groups of self-similarities of ergodic ows; see also [3], [13], [24], [25] for a recent contribution to other aspects of the problem of self-similarity of ergodic ows. From this point of view the subclass of so called GAG ows [17]1
of the class of Gaussian ows is especially attractive since self-similarities appear there as natural invariants, see (1.1) below. By de nition, GAG ows are those Gaussian ows whose ergodic self-joinings remain Gaussian. All Gaussian ows with simple spectrum are GAG ows [17]. If Tσ = (Tσ
t)t∈R denotes the Gaussian
ow determined by a nite positive (continuous) measure σ on R+ and the ow is
GAG then
(1.1) I(Tσ)is equal to the (multiplicative) group −I(σ) ∪ I(σ), where I(σ) = {s ∈ R∗
+ : σs ≡ σ}and σs = (Rs)∗(σ)denotes the image of σ via
the map Rs: x 7→ sx [17]. Recall that −1 is always a scale of self-similarity for a
Gaussian ow.
2010 Mathematics Subject Classi cation. Primary 37A10, 60G15; Secondary 43A05, 43A46.
1In [15] as well as in [17] only Gaussian automorphisms are considered, however all results can
be rewritten for Gaussian ows.
c
XXXX American Mathematical Society
In this note we focus on the problem of self-similarities in some subclasses of simple spectrum Gaussian ows. We rst recall already known results. Classically, if σ is concentrated on an additively Q-independent Borel set A ⊂ R+ then the
Gaussian ow Tσ has simple spectrum, see [2]. Moreover, the subgroup H :=
I(Tσ
) ∩ R∗+ is an additively Q-independent set. Indeed, suppose that H is not an
additively Q-independent set. That is, for some distinct h1, . . . , hm∈ H we have
(1.2) m X i=1 kihi= 0 with ki∈ Z, i = 1, . . . , m and m X i=1 ki2> 0.
Denote by H0 ⊂ H the multiplicative subgroup generated by h1, . . . , hm. Since
H0 ⊂ I(Tσ), we have σh ≡ σ for h ∈ H0, thus the Borel set B = Th∈H0hAhas
full σ-measure, is Q-independent, and is literally H0-invariant. Take any non-zero
x ∈ B. Then the elements hix ∈ B, i = 1, . . . , m, are distinct. Now, (1.2) yields m X i=1 ki(hix) = x m X i=1 kihi= 0,
so B is not independent, a contradiction. On the other hand, in [6], it is shown that whenever a countable group H ⊂ R∗
+ satis es:
(1.3) For each polynomial P ∈ Q[x1
, . . . , xm]if there is
a collection of distinct elements h1, . . . , hm in H such that
P (h1, . . . , hm) = 0then P ≡ 0,
then there exists a probability σ concentrated on a Borel Q-independent set such that I(Tσ) = −H ∪ H. It is not di cult to see that the condition (1.3) is equivalent
to saying that H is an additively Q-independent set.
Theorem 1.1 ([6]). Assume that G = −H ∪ H, where H ⊂ R∗
+ is a countable
multiplicative subgroup. Then G can be realized as I(Tσ) for a Gaussian ow
whose spectral measure σ is concentrated on a Borel Q-independent set if and only if H is an additively Q-independent set.
Note that for H cyclic generated by s ∈ R+, the Q-independence condition
is equivalent to saying that s is transcendental. Hence, by Theorem 1.1, a real number s can be realized as a scale of self-similarity of a Gaussian ow whose spectral measure is concentrated on a Q-independent Borel set if and only if s is transcendental.
On the other hand, there are no restrictions on H in the class of all Gaussian ows having simple spectrum.
Theorem 1.2 ([4]). For each countable subgroup H ⊂ R∗
+ there exists a simple
spectrum Gaussian ow Tσ such that I(Tσ) = −H ∪ H.
Note that, in particular, the above result of Danilenko and Ryzhikov brings the positive answer to the open problem [14] of existence of Gaussian ows Tσ
with simple spectrum such that σ is not concentrated on a Q-independent set; indeed, whenever H is not an additively Q-independent set, by Theorem 1.1, the spectral measure σ resulting from Theorem 1.2 cannot be concentrated on a Borel Q-independent set. See also [3] for constructions of Gaussian ows with zero entropy and having uncountable groups of self-similarities.
Our aim is to continue investigations on realization of countable subgroups as the groups of self-similarities in further restricted classes of Gaussian ows whose
spectral measures are classical from the harmonic analysis point of view. Recall some basic notions. For every s ∈ R let ξs: R → S1be given by ξs(t) = exp(2πist).
A nite positive Borel measure σ on R is called Kronecker if for each f ∈ L2
(R, σ), |f | = 1 σ-a.e., there exists a sequence (tn) ⊂ R, tn → ∞, such that
(1.4) ξtn→ f in L
2
(R, σ).
Each measure σ concentrated on a Kronecker set [12], [18] is a Kronecker measure. Indeed, Kronecker sets are compact subsets of R on which each continuous func-tion of modulus one is a uniform limit of characters. Kronecker sets are examples of Q-independent sets [18]. In general, as shown in [15], a Kronecker measure is concentrated on a Borel set which is the union of an increasing sequence of Kro-necker sets, hence a KroKro-necker measure is concentrated on a Borel Q-independent set, and the restriction on H in Theorem 1.1 applies. This turns out to be the only restriction as the main result of the note shows.
Theorem 1.3. Assume that G = −H ∪ H, where H ⊂ R∗
+ is a countable
multi-plicative subgroup. Then G can be realized as I(Tσ) 2 for a Gaussian-Kronecker
ow if and only if H is an additively Q-independent set. In particular, h ∈ R+
can be a scale of self-similarity for a Gaussian-Kronecker ow if and only if h is transcendental.
An extremal case when two dynamical systems are non-isomorphic is the dis-jointness in the Furstenberg sense [7], see also [9], [11], [14], [23] for disdis-jointness results in ergodic theory. We would like also to emphasize that the notion of dis-jointness turned out to be quite meaningful in the problem of non-correlation with the Möbius function of sequences of dynamical origin [1]: we need that an automor-phism T has the property that Tp and Tq are disjoint for any two dierent primes.
In connection with that we will prove the following. Theorem 1.4. Assume that Tσ = (Tσ
t)t∈R is a Gaussian-Kronecker ow. If s ∈
Q\{±1} then Tsσis disjoint from T1σ. For every Gaussian-Kronecker automorphism
T : (X, B, µ) → (X, B, µ) the iterations Tn, Tm are disjoint for any two distinct
natural numbers n, m.
If s is irrational then there exists a Gaussian-Kronecker ow Tσ such that Tσ s
and Tσ
1 have a non-trivial common factor.
An importance of Kronecker measures in ergodic theory follows from the follow-ing remarkable result of Foia³ and Stratila [5] (see also [2], and remarks on that result in [15] and [21]):
(1.5)
If (St)t∈R is an ergodic ow of a standard probability
Borel space (Y, C, ν), f ∈ L2(Y, C, ν) is real and the spectral measure
σf of f is the symmetrization of a Kronecker measure,
then the (stationary) process (f ◦ St)t∈R is Gaussian.
In [15], any measure σ satisfying the assertion (1.5) of Foia³-Stratila theorem is called an FS measure. Each Kronecker measure is a Dirichlet measure3[18], but as 2In a sense, we can also control the ows Tσsfor s /∈ −H ∪ H; we will prove their disjointness from Tσ, see the proof of this theorem.
3A probability Borel measure σ on R is Dirichlet, if (1.4) is satis ed for f = 1. From the
dynamical point of view, Dirichlet measures correspond to rigidity: a ow T is rigid if Ttn→ Id for some tn→ ∞.
shown in [15], there are FS measures which are not Dirichlet measures (see Figure 1). Moreover, in [21], it is announced that each continuous measure concentrated on
Kronecker measures
FS measures
Dirichlet measures
Figure 1. Dierent classes of measures
an independent Helson4 set is a Kronecker measure (for some examples in [21],
the resulting Gaussian ows have no non-trivial rigid factors). We will strenghten Theorem 1.2 to the following result.
Theorem 1.5. Any symmetric countable group G ⊂ R∗ can be realized as the
group of self-similarities of a simple spectrum Gaussian ow Tσ with σ being an
FS measure.
In particular, in connection with the forementioned question from [14], there is an FS measure for which the Gaussian ow has simple spectrum but σ is not concentrated on a Q-independent set. These are apparently the rst examples of FS measures which are not concentrated on Q-independent Borel sets but yield Gaussian ows with simple spectrum (cf. [15] and [21]).
At the end of the note we will discuss self-similarity properties of Gaussian ows arising from a typical measure or from the maximal spectral types of a typical ow (cf. the disjointness results from [4]).
Theorem 1.6. Assume that 0 ≤ a < b. For a typical σ ∈ P([a, b]) the ow Tσ is
Gaussian-Kronecker such that for each |r| 6= |s| the ows Tσr and Tσs are disjoint.
In particular I(Tσ) = {±1}.
For a typical ow T of a standard probability Borel space (X, B, µ), for its maximal spectral type σT we have: TσT|R+ has simple spectrum and for |r| 6= |s|
the ows T(σT|R+)r and T(σT|R+)s are disjoint. In particular I(TσT|R+) = {±1}.
2. Notation and basic results
Assume that T = (Tt)t∈R is a measurable5 measure-preserving ow acting on
a standard probability Borel space (X, B, µ). It then induces a (continuous) one-parameter group of unitary operators acting on L2(X, B, µ) by the formula T
tf =
f ◦ Tt. By Bochner's theorem, the function t 7→ RXTtf · f dµ determines the so
called spectral measure σf of f for whichσbf(t) = R
XTtf · f dµ, t ∈ R. Usually, one
only considers spectral measures of f ∈ L2
0(X, B, µ), that is, of elements with zero
mean (the spectral measure of the constant function c is equal to |c|2δ
0). Then
σf is a nite positive Borel measure on R. Among spectral measures there are
maximal ones with respect to the absolute continuity relation. Each such maximal
4A ⊂ R is called Helson if for some δ > 0 and each complex Borel measure κ concentrated on
Athe supt∈R R
Re
2πitxdκ(x)
is bounded away from the δ-fraction of the total variation of κ.
5Measurability means that for each f ∈ L2(X, B, µ)the map t 7→ f ◦ T
measure is called a maximal spectral type measure and, by some abuse of notation, it will be denoted by σT. We refer the reader to [11] and [14] for some basics about
spectral theory of unitary representations of locally compact Abelian groups in the dynamical context.
Assume that T is ergodic and let S = (St)t∈R be another ergodic ow (acting
on (Y, C, ν)). Any probability measure ρ on (X × Y, B ⊗ C) which is (Tt× St)t∈R
-invariant and has marginals µ and ν respectively, is called a joining of T and S. If, additionally, the ow ((Tt×St)t∈R, ρ)is ergodic then ρ is called an ergodic joining6.
The ergodic joinings are extremal points in the simplex of all joinings. If the set of joinings is reduced to contain only the product measure then one speaks about disjointness of T and S [7] and we will write T ⊥ S. Similar notions appear when one considers automorphisms. Note that whenever for some t 6= 0, Tt ⊥ St then
T ⊥ S. Note also that whenever
(2.1) T is weakly mixing then T ⊥ S if and only if T1⊥ S1.
Indeed, if T1 6⊥ S1 then there exists an ergodic joining ρ between them dierent
than the product measure. Then, ρ◦(Tr×Sr)for 0 ≤ r < 1 has the same properties.
By disjointness of T and S, R1
0 ρ ◦ (Tr× Sr) dr = µ ⊗ ν. But T1 is weakly mixing,
so µ ⊗ ν is an ergodic joining of T1 and S1, and therefore ρ ◦ (Tr× Sr) = µ ⊗ ν. We
refer the reader to [9] for the theory of joinings in ergodic theory.
A ow T is called Gaussian if there is a T -invariant subspace H ⊂ L2
0(X, B, µ)
of the zero mean real-valued functions such that all non-zero variables from H are Gaussian and the smallest σ-algebra making all these variables measurable equals B. A Gaussian ow is ergodic if and only if the maximal spectral type on H is continuous (and then T is weakly mixing). Since Gaussian variables are real, it is not hard to see that their spectral measures are symmetric, that is, for f ∈ H, σf
is invariant under the map R−1 : x 7→ −x.
A standard way to obtain a (weakly mixing) Gaussian ow is to start with a nite positive continuous Borel measure σ on R+. Consider the symmetrization
e
σ = σ + (R−1)∗σ7. We let V = (Vt)t∈R denote the one-parameter group of unitary
operators on L2
(R,eσ)de ned by Vt(f )(x) = e
2πitxf (x). Then the correspondence
(2.2) f (x) 7→ f (−x)
yields the unitary conjugation of V and its inverse. Let (X, B, µ) be a Gaussian probability space, that is, a standard probability space together with an in nite dimensional, closed, real and B-generating subspace H ⊂ L2(X, B, µ) whose all
non-zero variables are Gaussian. We then consider H + iH, so called complex Gaussian space, and de ne an isomorphic copy of V on it. It is then standard to show (see e.g. [17], Section 2) that V has a unique extension to a (measurable) ow Tσ = (Tσ
t) of (X, B, µ) for which UTσ
t |H+iH = Vt, t ∈ R. By the same token, the
correspondence (2.2) extends to an isomorphism of (X, B, µ) which conjugates the Gaussian ow and its inverse (Tσ
−t)t∈R.
A Gaussian ow Tσis called Gaussian-Kronecker (FS resp.) if σ is a continuous
Kronecker (FS resp.) measure. Following [17], a Gaussian ow Tσ (with the 6If T = S then we speak about self-joinings.
7In general, when f is a measurable map from (X, B) to (Y, C) and κ is a probability measure
Gaussian space H) is called GAG if for each its ergodic self-joining η the space {f (x) + g(y) : f, g ∈ H}
consists solely of Gaussian variables (the ow (Tσ t × T
σ
t)t∈Ris then a Gaussian ow
as well). We have [17]
Gaussian -Kronecker flows
FS-flows
simple spectrum flows GAG flows
Figure 2. Dierent subclasses of GAG ows
For all these classes of ows we have that if Tσis in the class, so is Tσsfor s 6= 0.
In general, Gaussian ows given by equivalent measures are isomorphic. It fol-lows from [17] that any isomorphism between a GAG ow Tσand another Gaussian
ow Tν is entirely determined by a unitary isomorphism of restrictions of the
uni-tary actions (Tσ
t)t∈Rand (Ttν)t∈Rto their Gaussian subspaces. That is, in the GAG
situation, Tσ are Tν are isomorphic if and only if σ ≡ ν. If we apply that to σ and
σsfor s ∈ R+ we will immediately get (1.1) to hold (in the GAG case).
We will now prove the following.
Proposition 2.1. Assume that Tσ is GAG. Fix s 6= 0. Then the sets of
self-joinings of Tσ and of self-joinings of Tσ
s are the same. (Hence ergodic self-joinings
are also the same.) In particular, the factors and the centralizer of the ow and of the time s-automorphism are the same.
Proof. This follows from the proof of Theorem 6.1 in [10] which asserts that such an equality of the sets of self-joinings takes place whenever each ergodic self-joining of the ow is an ergodic self-joining for the time-s automorphism. In the GAG case, by de nition, such ergodic joinings for the ow Tσ are Gaussian joinings, so they
are automatically ergodic for the Tσ
s [17].
Corollary 2.2. Assume that Tσ is GAG. Then Tσ
s is a GAG automorphism for
each s 6= 0.
We will also make use of the following results.
Theorem 2.3 ([17]). Assume that Tσis GAG and let Tη be an arbitrary Gaussian
ow. Then Tσ⊥ Tη if and only if
e
σ ⊥eη ∗ δr for each r ∈ R.
Proposition 2.4 ([15]). If σ1 and σ2are measures with the FS property and Tσ1⊥
Tσ2 then σ = 1
3. Auxiliary lemmas
Given a compact subset X ⊂ R denote by P(X) the set of all Borel probabil-ity measures concentrated on X endowed with the usual weak topology which is compact and metrizable: if {fn: n ≥ 1} is a dense set in C(X) then
(3.1) d(σ, η) = ∞ X n=1 1 2n R fndσ −R fndη 1 + R fndσ −R fndη
de nes a metric compatible with the weak topology. Denote U(X) = {f ∈ C(X) : |f | = 1} which is a closed subset of C(X) in the uniform topology, in particular U (X)is a Polish space.
Lemma 3.1. Assume that X = [a, b]. Let {h0, h1, . . . , hm} ⊂ R∗be a Q-independent
set. Then for each f ∈ USm
j=0hjX and ε > 0 (3.2) n Af,ε(h1, . . . , hm) := σ ∈ P ([a, b]) : (∃t ∈ R) kf − ξtkL2(R, 1 m+1 Pm j=0σhj) < ε o
is open and dense in P([a, b]).
Proof. The set Af,ε(h1, . . . , hm)is clearly open, so we need to show its denseness in
P(X). Since discrete measures with a nite number of atoms form a dense subset of P(X)we take ν = PNs=1asδys with ys∈ [a, b], as> 0, s = 1, . . . , N and P
N s=1as=
1 and x δ > 0. All we need to show is to nd a subset {x1, . . . , xN} ⊂ [a, b]such
that |xs− ys| < δ for s = 1, . . . , N and such that the set
L :=
m
[
j=0
{hjx1, . . . , hjxN} is Q-independent.
Indeed, in this case by Kronecker's theorem, the set L is a nite Kronecker set, so the measure 1 m+1 Pm j=0 PN s=1asδxs hj
is Kronecker, whence belongs to Af,ε(h1, . . . , hm)
and it δ-approximates ν. To show that x1, . . . , xN can be selected so that L is
Q-independent, consider the algebraic varieties of the form (z1, . . . , zN) ∈ X×N : m X j=0 N X s=1 kjshjzs= 0 for some non-zero integer matrix (kjs). Since
m X j=0 N X s=1 kjshjzs= N X s=1 m X j=0 kjshj zs and Pm
j=0kjshj 6= 0 whenever (k0s, . . . , kms) 6= (0, . . . , 0) (and there are such
vectors since the matrix (kjs) is not zero), each such variety has N-dimensional
Lebesgue measure zero. Since there are only countably many such varieties in-volved, we may discard the union S of them from [a, b]×N. Now, each choice of
(x1, . . . , xN)from (y1− δ, y1+ δ) × . . . × (yN − δ, yN + δ) \ S satis es our
Lemma 3.2. Given H ⊂ R∗
+ a countable subset which is a Q-independent set, the
set of continuous (Kronecker) measures σ ∈ P([a, b]) for which the measure
(3.3) X
h∈H
ahσh is a Kronecker measure (on R)
for each choice of ah≥ 0, Ph∈Hah= 1, is a Gδ and dense subset of P([a, b]).
Proof. Denote by Pc([a, b])the set of continuous measures which is a Gδ and dense
subset of P([a, b]). Let H = {h0, h1, h2, . . .}. For every m ≥ 0 x a countable dense
familyng(m)i : i ≥ 1o⊂ U (Sm
i=0hi[a, b]). Then, by Lemma 3.1, the set
Pc([a, b]) ∩ ∞ \ m=1 ∞ \ i=1 ∞ \ p=1 Ag(m) i ,1p (h1, . . . , hm)
is Gδ and dense in P([a, b]) and it remains to show that this is precisely the set of
measures satisfying (3.3). Indeed, given m ≥ 1, the set Km(H) := Pc([a, b]) ∩ ∞ \ i=1 ∞ \ p=1 Ag(m) i , 1 p (h1, . . . , hm)
is precisely the set of continuous Kronecker measures σ ∈ P([a, b]) such that the measure 1
m+1
Pm
i=0σhi is a Kronecker measure (on the real line). Moreover, each
measure absolutely continuous with respect to a Kronecker measure is also a Kro-necker measure [15]. Therefore the set Km(H) is equal to the set of all Kronecker
measures σ ∈ P([a, b]) such that Pm
i=0biσhi is Kronecker for arbitrary choice of
bi ≥ 0, Pmi=0bi= 1. Finally, for each m ≥ 1,
1 Pm i=0ahi m X i=0 ahiσhi 1 m + 1 m X i=0 σhi,
so if for each m ≥ 1 the measure 1 m+1
Pm
i=0σhiis Kronecker, so is Ph∈Hahσh.
Remark 3.3. The idea of the above proofs is taken from a letter that has been sent to us by T.W. Körner. In this letter, T.W. Körner shows that given a transcendental number h ∈ R, for a typical (in the Hausdor metric) closed subset K ⊂ [a, b] the set K ∪ hK is Kronecker and uncountable. The proofs are the same since nite sets are dense in the Hausdor metric and if h is transcendental then given distinct y1, . . . , yN ∈ [a, b] and δ > 0 we can nd qi ∈ Q so that for xi := h2iqi we have
|xi− yi| < δ for i = 1, . . . , N and clearly the set {x1, . . . , xN, hx1, . . . , hxN} is
Q-independent. It only remains to notice that uncountable closed subsets are typical in the Hausdor metric.
Note also that using the proofs of Lemmas 3.1 and 3.2, given H ⊂ R∗
+ a
count-able multiplicative subgroup which is additively Q-independent, we obtain that a typical (with respect to the Hausdor distance) closed subset K ⊂ [a, b] has the property that for each nite subset C ⊂ H the set Sh∈ChK is Kronecker, so the
set Sh∈HhK is a Q-independent Fσ-set.
We will also need the following compact Q-independent set version of Lemma 3.2. Lemma 3.4. Assume that K ⊂ R is a compact uncountable set. Assume that H ⊂ R∗
that the set Sh∈HhK is Q-independent. Then the set of continuous (Kronecker)
measures σ concentrated on K for which the measure
(3.4) X
h∈H
ahσh is a Kronecker measure
for each choice of ah≥ 0, Ph∈Hah= 1, is a Gδ and dense subset of P(K).
Proof. This follows from the proofs of Lemmas 3.1 and 3.2, where in addition the proof of Lemma 3.1 is simpli ed; indeed, for any choice of {y1, . . . , yN} ⊂ K the
set Sm
j=0{hjy1, . . . , hjyN}is Q-independent by assumption (so we may take xi= yi
for i = 1, . . . , N).
Remark 3.5. For any non-trivial compact interval [a, b] ⊂ R denote by P[a,b]
c (R)
the set of measures ν ∈ Pc(R) such that ν([a, b]) > 0. Since the map Pc(R) 3
ν 7→ ν([a, b]) ∈ R is continuous, the set Pc[a,b](R) is open and dense in Pc(R).
Let us consider the map ∆ = ∆[a,b] : P[a,b]
c (R) → Pc([a, b]) such that ∆(ν) is
the conditional probability measure ν( · |[a, b]). This map is continuous and the preimage of any dense subset of Pc([a, b]) is dense in P
[a,b]
c (R). Indeed, let A ⊂
Pc([a, b])be dense and take any ν ∈ P [a,b]
c (R). Then there exists a sequence (eνn)n≤1
in A such thatνen → ∆(ν)weakly. For every n ≥ 1 de ne νn ∈ P
[a,b]
c (R) so that
the restriction of νn to [a, b] is ν([a, b])eνn and the measures νn and ν coincide on
R \ [a, b]. Then ∆(νn) =eνn ∈ A and νn → ν weakly. Consequently, the preimage
∆−1Aof any Gδ dense subset A ⊂ Pc([a, b])is Gδ dense in P [a,b] c (R).
Before we prove a certain disjointness property of Kronecker measures, we will need the following general observation.
Lemma 3.6. Let (X, B) be a standard Borel space and let ϕ : X → X be a measurable map. Let σ be a nite positive continuous Borel measure on X such that the map ϕ : (X, σ) → (X, ϕ∗σ)is almost everywhere invertible. Assume that σ({x ∈
X : ϕ(x) = x}) = 0 and that the measures σ and ϕ∗σ are not mutually singular.
Then there exists a measurable set A ∈ B such that σ(A) > 0, σ(A ∩ ϕ−1A) = 0
and the measures σ and ϕ∗σrestricted to A are equivalent.
Proof. By assumption, there exists Y ∈ B such that σ(Y ) > 0 and the measures σ and ϕ∗σrestricted to Y are equivalent. It follows that if A ∈ B, A ⊂ Y , σ(A) > 0,
then the measures σ and ϕ∗σrestricted to A are also equivalent.
Case 1. Suppose that there exists B ∈ B such that B ⊂ Y and σ(B \ ϕ(B)) > 0. Set A := B \ ϕ(B). Then σ(A) > 0 and A ∩ ϕ−1A = (B \ ϕ(B)) ∩ (ϕ−1B \ B) = ∅.
Since A ⊂ B ⊂ Y , our claim follows.
Case 2. Suppose that for every B ∈ B with B ⊂ Y we have σ(B \ ϕ(B)) = 0. As σ and ϕ∗σrestricted to Y are equivalent, it follows that
(3.5) 0 = ϕ∗σ(B \ ϕ(B)) = σ(ϕ−1B \ B) for every B ⊂ Y.
We now show that there exists A ∈ B such that A ⊂ Y , σ(A) > 0 and σ(A∩ϕ−1A) =
0, which gives our assertion. Suppose that, contrary to our claim, for every A ∈ B with A ⊂ Y the condition σ(A) > 0 implies σ(A ∩ ϕ−1A) > 0. It follows that
Indeed, otherwise for some B as above, A := B \ ϕ−1(B) ⊂ Y would be of positive
σ-measure and since
(B \ ϕ−1B) ∩ ϕ−1(B \ ϕ−1B) = (B \ ϕ−1B) ∩ (ϕ−1B \ ϕ−2B) = ∅, and we would get σ(A ∩ ϕ−1A) = 0, a contradiction.
Now, (3.5) combined with (3.6) gives σ(B4ϕ−1B) = 0 for every B ∈ B with
B ⊂ Y. It follows that ϕ(x) = x for σ-a.e. x ∈ Y , contrary to assumption. For any real s let θs : R → R, θs(t) = t + s. Recall that for every n ∈ Z and
z1, z2∈ S1we have
(3.7) |zn
1 − z2n| ≤ |n||z1− z2|.
Lemma 3.7. Let σ be a continuous Kronecker measure on R. Then for every s ∈ Q∗\ {1} and r ∈ R we have σ ⊥ σs∗ δr.
Proof. Suppose that, contrary to our claim, there exists s ∈ Q∗\ {1} and r ∈ R
such that σ 6⊥ σs∗ δr. Let ϕ := θr◦ Rs. Then ϕ : R → R is an invertible map with
one xed point and σs∗ δr= ϕ∗σ. By Lemma 3.6, there exists a Borel set A0⊂ R
such that σ(A0) > 0, σ(A0∩ ϕ−1A0) = 0 and the measures σ, ϕ∗σ restricted to
A0 are equivalent. Thus σ(ϕ−1A0) > 0. Let A1, A2⊂ A0 be disjoint Borel subsets
such that σ(ϕ−1A
1) > 0and σ(ϕ−1A2) > 0.
Let s = q/p with p and q relatively prime integer numbers. Choose z0∈ S1such
that zq
06= 1. Let us consider the measurable map f : R → S1 such that f(x) = z0
if x ∈ ϕ−1A
2 and f(x) = 1 otherwise. Since σ is a Kronecker measure, there
exists a sequence (tn)n∈N of real numbers such that ξtn → f in L 2 (R, σ). Thus ξtn◦ ϕ −1→ f ◦ ϕ−1 in L2 (R, ϕ∗σ). Since gn0(x) := χA0(x) |exp(2πitnx) − 1| ≤ |ξtn(x) − f (x)| gn1(x) := χA1(x) exp(2πitns−1(x − r)) − 1 ≤ ξtn(ϕ −1x) − f (ϕ−1x) gn2(x) := χA2(x) exp(2πitns−1(x − r)) − z0 ≤ ξtn(ϕ −1x) − f (ϕ−1x) , it follows that (g0
n)tends to zero in measure σ and the sequences (gn1), (gn2) tend
to zero in measure ϕ∗σ. As σ ≡ ϕ∗σon A0 and A1, A2 ⊂ A0, the sequences (g1n),
(g2
n)tend to zero in measure σ, as well. Fix
(3.8) 0 < ε < |z
q 0− 1|
2(|p| + |q|). Then there exist measurable sets A0
k ⊂ Ak, k = 0, 1, 2 and n ∈ N such that for
k = 0, 1, 2
σ(Ak\ A0k) <
1
4min(σ(A1), σ(A2))and g
k
n(x) < εfor all x ∈ A 0 k.
Therefore for k = 1, 2 we have σ(Ak\ A00) ≤ σ(A0\ A00) <
1
4σ(Ak)and σ(Ak\ A
0 k) <
1 4σ(Ak),
so σ(A0
0∩ A0k) > σ(Ak)/2 > 0. Choose two real numbers x1 ∈ A00 ∩ A01 and
x2∈ A00∩ A02. Then |exp(2πitnx1) − 1| = gn0(x1) < ε, exp(2πitn p q(x1− r)) − 1 = g1n(x1) < ε, |exp(2πitnx2) − 1| = gn0(x2) < ε, exp(2πitn p q(x2− r)) − z0 = g2n(x2) < ε. In view of (3.7), |exp(2πitnp x1) − 1| < |p|ε, |exp(2πitnp(x1− r)) − 1| < |q|ε, |exp(2πitnp x2) − 1| < |p|ε, |exp(2πitnp(x2− r)) − z0q| < |q|ε. Hence |exp(2πitnpr) − 1| < (|p| + |q|)ε, |exp(2πitnpr)z0q− 1| < (|p| + |q|)ε, so |1 − z0q| < 2(|p| + |q|)ε, contrary to (3.8).
Let us now consider the space P(R) of all Borel probability measures on R endowed with the weak topology.
By supp(σ) we always mean the topological support of the measure σ. Let us recall that
if σ ∈ P(R) has supp(σ) = R
then the set {ν ∈ P(R): ν σ} is dense in P(R). (3.9)
Denote by Pc(R) the set of all continuous members of P(R) (this is a Gδ and dense
subset of P(R)).
The proof of the lemma below is a slight modi cation of the proof of Lemma 3.1 from [4].
Lemma 3.8. The set
S = {σ ∈ Pc(R) : σs⊥ σ ∗ δt for each 1 6= s ∈ R∗, t ∈ R}
is Gδ and dense in P(R).
Proof. Denote by I the family of open subset of R which are nite unions of open intervals. Recall that for two measures σ, ν ∈ P(R)
(3.10) σ ⊥ ν ⇐⇒ ∀n∈N∃O∈I σ(O) < 1/n and ν(O) > 1 − 1/n.
For any compact rectangle I × J ⊂ (R∗\ {1}) × R denote by V(I × J) the set of
all nite covers of I × J by compact rectangles contained in (R∗
\ {1}) × R. Notice that for each open subset O ∈ I the map
(3.11) Pc(R) × R∗× R 3 (σ, s, r) 7→ σs∗ δr(O) ∈ R
is continuous. Therefore, given a compact rectangle F ⊂ (R∗\ {1}) × R and an
open subset O ∈ I the map fF,O: Pc(R) 3 σ 7→ σ(O), max (s,r)∈Fσs∗ δr(O) ∈ R2
is continuous. Let e S = \ I631 \ J \ n∈N [ κ∈V(I×J ) \ F ∈κ [ O∈I fF,O−1 ((1 − 1/n, ∞) × (−∞, 1/n)) ,
where I and J run over closed intervals with rational endpoints. Then Seis a Gδ
set.
We claim that S = Se . Indeed, let σ ∈ S. Let I 63 1 and J ⊂ R be compact intervals and n ∈ N. By assumption and (3.10), for every (s0, r0) ∈ I × J there
exists an open set Os0,r0 ∈ I such that
σ(Os0,r0) > 1 − 1/n and σs0∗ δr0(Os0,r0) < 1/n.
Since the map (3.11) is continuous, there exist open rectangles U0
s0,r0⊂ Us0,r0 ⊂ R 2
such that (s0, r0) ∈ Us00,r0 and a compact rectangle Fs0,r0 ⊂ (R
∗\{1})×R satisfying
Us0
0,r0 ⊂ Fs0,r0 ⊂ Us0,r0 such that
σs∗ δr(Os0,r0) < 1/n for all (s, r) ∈ Us0,r0.
Since I × J is compact and {U0
s,r: (s, r) ∈ I × J }is its open cover, there exists a
nite cover κ := {Fs1,r1, . . . , Fsk,rk} of I × J. It follows that
fFsj ,rj,Osj ,rj(σ) ∈ (1 − 1/n, ∞) × (−∞, 1/n) for all j = 1, . . . , k,
thus σ ∈Se.
Suppose that σ ∈Seand x s0∈ R∗\ {1}, r0∈ R and n ∈ N. Next choose I 63 1
and J ⊂ R compact intervals such that (s0, r0) ∈ I × J. By assumption, there
exists a nite cover κ ∈ V(I × J) such that for every F ∈ κ there exists OF ∈ I
with
σ(OF) > 1 − 1/n and σs∗ δr(OF) < 1/n for all (s, r) ∈ F.
Choosing F ∈ κ for which (s0, r0) ∈ F and applying (3.10) we have that σ and
σs0∗ δr0 are orthogonal, so σ ∈ S.
It remains to show that S is dense. To this end we use the proof of Proposition 3.4 in [4]. Namely, in this proposition there is a construction of a weakly mixing ow T such that for a certain sequence of real numbers uk → ∞we have: for each
l ∈ N
(3.12) T−duk → 10
−l for d = 1 − 10−l and
(3.13) T−cuk→ 0 uniformly in c ∈ [1, 10 l]
(the convergence takes place in the weak operator topology). It follows that
(3.14) σTd ⊥ σTc∗ δt
for all t ∈ R; indeed, (3.12) and (3.13) mean respectively ξuk→ 10 −l weakly in L2 (R, σTd), and ξuk→ 0 weakly in L 2 (R, σTc).
It is easy to see that the latter condition implies ξuk → 0 weakly in L
2
(R, σTc∗ δt)
Now, in view of (3.14), σT ⊥ σTc/d∗ δt/d, and since in (3.13) c can be replaced by
−c, it follows that σT ∈ S. It is also clear that S is closed under taking absolutely
continuous measures. Since supp σT = R8, the result follows from (3.9).
Recall also the following basic observation.
Lemma 3.9. Let s = (sj)j≥1 be a sequence of positive numbers and let g = (gj)j≥1
be a sequence of uniformly bounded continuous functions. Then the set Ws,g=ν ∈ P(R): (∃ tn→ ∞) (∀ j ≥ 1) ξsjtn→ gj weakly in L
2
(R, ν) is Gδ in P(R).
Proof. Let (fm)m≥1be a sequence of continuous functions on R uniformly bounded
by 1, which is linearly dense in L2
(R, ν) for every ν ∈ P(R). Set R(n, ε) = µ ∈ P(R) : X m,j≥1 1 2m+j Z R e2πisjnx− g j(x) fm(x) dµ(x) < ε .
The set R(n, ε) is open. To complete the proof it su ces to notice that Ws,g= \ Q3ε>0 \ m≥1 [ n≥m R(n, ε). Lemma 3.10. Let H ⊂ R∗
+ be a countable multiplicative subgroup. Then for a
typical ν ∈ P(R) the measure η := Ph∈Hahνh (with ah > 0 and Ph∈Hah = 1)
yields a Gaussian ow Tη|R+ with simple spectrum.
Proof. Set G = −H ∪ H and let H = {si : i ≥ 0}(s0= 1). In [4], Danilenko and
Ryzhikov constructed a rank-1 ow T preserving a σ- nite measure µ (the ow acts on (X, B, µ)) such that if σ = σT denotes its maximal spectral type on L2(X, B, µ)
then the Gaussian ow
(3.15) T (Pi≥1
1
2iσsi)|R+ has simple spectrum.
To prove this, they used the following properties of T : a) T√
2s∈ W CP (Ts)9for each s ∈ H,
b) 1 qI +
q−1
q Ts∈ W CP (Ts)for each s ∈ H and q ∈ N,
c) for each nite sequence s1 < s2 < · · · < sk of elements of H and each
1 ≤ l0≤ kthere exists tj→ ∞such that
(i) Ttjsj → 1 2kIif 1 ≤ l ≤ k, l 6= l0, (ii) Ttjsl0 → 1 2kTsl0.
Notice that the conditions a), b) and c) can be expressed as follows in terms of weak convergence of continuous and bounded functions in L2
(R, σ): a') for each s ∈ H there exists a sequence nk → ∞such that
ξsnk → ξ√2s,
8This fact is well known for Z-actions, e.g. [20], Chapter 3, and can be easily rewritten using
special representation of ows. See also the proof of Theorem A in [22].
9An operator Q belongs to the weak closure of powers WCP(R) if for an increasing sequence
b') for each s ∈ H and q ∈ N there exists a sequence nk → ∞such that ξsnk→ 1 q+ q − 1 q ξs,
c') for each nite sequence s1 < s2 < · · · < sk of elements of H and each
1 ≤ l0≤ kthere exists tj→ ∞such that
(i) ξtjsj → 1 2k if 1 ≤ l ≤ k, l 6= l0, (ii) ξtjsl0 → 1 2kξsl0.
The arguments used in the proof of Theorem 4.4 in [4] show that for each continuous probability measure σ on R conditions a'), b') and c') imply the simplicity of spectrum of the ow T(P
k≥1 1
2kσsk)|R+. Moreover, by Lemma 3.9, the set of measures
ν ∈ P(R) satisfying these conditions is Gδ. We will show now that it is also dense
in P(R). Notice that conditions a'), b') and c') hold also in L2
(R, ν) for any ν σ. Since σT is the maximal spectral type of a rank-1 in nite measure-preserving ow
T, the Gelfand spectrum of the corresponding Koopman representation is equal to R. It follows that the topological support of σT is full and therefore the result
follows from (3.9).
4. Proofs of theorems
Proof of Theorem 1.3. (based on Lemmas 3.2 and 3.8.) Using these two lemmas, for a typical (continuous, Kronecker) measure σ ∈ P([a, b]) we have (with ah> 0,
and Ph∈Hah= 1)
−H ∪ H ⊂ I(Tη),
where η := Ph∈Hahσh is a Kronecker measure and moreover
(4.1) σs⊥ σ ∗ δt
for each non-zero real s 6= 1 and arbitrary t ∈ R. All we need to show is that when s /∈ −H ∪ H then ηs6≡ η. However if s /∈ H then even more is true: η ⊥ ηs∗ δtfor
arbitrary t ∈ R and s /∈ {0, 1}. It follows that e
ηs⊥η ∗ δe t
for each s /∈ −H∪H and t ∈ R. In view of Theorem 2.3, it follows that Tηis disjoint
from Tηs (isomorphic to Tη
s) for s /∈ −H ∪ H. In particular, −H ∪ H = I(Tη)and
the result follows.
Proof of Theorem 1.3. (based on Lemma 3.4.) Given H ⊂ R∗
+ a multiplicative
subgroup which is an additively Q-independent set, in [6], there is a construction of a perfect compact set K such thatK :=b
S
h∈HhK is independent and forK :=e − bK ∪ bKthe following holds: (rK +t)∩ ee Kis countable whenever |r| /∈ H and t ∈ R is arbitrary. Using Lemma 3.4 nd a (continuous, Kronecker) measure σ ∈ P(K) such that η := Ph∈Hahσh is a Kronecker measure. Then η is concentrated onKb. All we need to show is that if |r| /∈ H, then the symmetrization of ηr is not equivalent
to the symmetrization of η. This is however clear, since the symmetrization of ηr
is a continuous measure concentrated on rKe. As in the previous proof we deduce that for s /∈ −H ∪ H we obtain disjointness of the corresponding ows. Proof of Theorem 1.4. First notice that directly from Lemma 3.7, it follows that whenever σ is a Kronecker measure then for each r1, r2∈ Q∗, r16= r2, we have
It follows thateσr1 ⊥σer2∗δtfor all t ∈ R, so by Theorem 2.3, the Gaussian-Kronecker
ows Tσr1 and Tσr2 are disjoint. In view of (2.1), it follows that Tσr1
1 ⊥ T σr2 1 , thus Tσ r1 ⊥ T σ r2.
Now suppose that T = Tσ : (X, B, µ) → (X, B, µ) is a Gaussian-Kronecker
automorphism, i.e. σ = σ0+ σ0 for a continuous Kronecker measure σ0∈ P(T).10
Denote by σ0 the image of σ
0 via the map T 3 z 7→ Arg(z)/2π ∈ [0, 1). Then σ0 is
a continuous Kronecker measure on R such that (ξ1)∗eσ
0 = σ and
e
σ0∗ δm ⊥σe0 for
all m ∈ N. Denote by H the Gaussian space of the ow Tσ0. Then the Koopman
operator of Tσ0
1 has simple spectrum on H and its spectral type is (ξ1)∗σe
0= σ, see
Appendix in [16]. Since the spectral type of ζ1(with respect to Tσ 0
1 ) is (ξ1)∗eσ
0= σ,
it follows that ζ1◦ (Tσ 0
1 )n, n ∈ Z, span the space H. Thus Tσ 0
1 is isomorphic to Tσ.
By the rst assertion of the theorem, it follows that Tn
σ is disjoint from Tσmfor any
pair of distinct natural numbers.
In order to prove the second part of the theorem note that if s is irrational then the set {1, s} is Q-independent, so by Lemma 3.2 we can nd a (continuous, Kronecker) measure σ ∈ P([a, b]) such that η := 1
2(σ + σs)is a Kronecker measure.
Since σs η and σs ηs the Gaussian-Kronecker ows Tη and Tηs have a
common non-trivial (Gaussian) factor. Its time one map is a common non-trivial factor of Tη
1 and T ηs
1 and it remains to notice that the Gaussian automorphism T ηs 1
is isomorphic to Tη
s.
Proof of Theorem 1.5. Let H = G ∩ R∗
+ and let (ah)h∈H be positive numbers such
that Ph∈Hah = 1. By Lemmas 3.8, 3.10 and Lemma 3.2 (applied to H = {1})
combined with Remark 3.5, there exists ν0 ∈ P
c(R) such that
(i) ν0
s⊥ ν0∗ δtfor all s ∈ R∗\ {1}and t ∈ R;
(ii) the Gaussian ow T(P
h∈Hahνs0)|R+ has simple spectrum
(iii) ν := ∆(ν0) ∈ P
c([a, b])is a Kronecker measure
(in fact, for a typical ν0∈ P
c(R) the properties (i)-(iii) hold). Since the conditions
(i) and (ii) hold also for any measure absolutely continuous with respect to ν, the Kronecker measure ν satis es (i) and (ii) as well. Therefore, setting σ := P
h∈Hahνs, by (ii), the Gaussian ow Tσhas simple spectrum. The same argument
as in the proof of Theorem 1.3 shows that (i) together with (ii) imply I(Tσ) =
−H ∪ H and Tνs ⊥ Tνr whenever |r| 6= |s|. Each Kronecker measure ν
h, h ∈ H
is an FS measure so, by Proposition 2.4, it follows that σ = Ph∈Hahνs is an FS
measure11, which completes the proof.
Proof of Theorem 1.6. The rst part follows from Lemma 3.8 along the same lines as the rst proof of Theorem 1.3 (for H = {1}).
In view of Corollary 2 in [16], a typical ow T has the SC property,12 which is
equivalent to the fact that TσT has simple spectrum. In particular, it implies that
TσT is GAG.
In order to prove that σT ⊥ (σT)s∗δr, s ∈ R∗\{1}, r ∈ R for a typical ow T we
follow the proof of Theorem 3.2 from [4] (using Lemma 3.8 and the existence of a
10T stands for {z ∈ C : |z| = 1}.
11We use here the elementary fact that the L2-limit of a sequence of Gausian variables remains
Gaussian.
12The SC property means that if we set σ = σT then for each n ≥ 2 the conditional measures
of the disintegration of σ⊗nover σ∗nvia the map Rn3 (x
1, . . . , xn) 7→ x1+ · · · + xn∈ R are
ow satisfying (3.14)). Since TσT is GAG for a typical ow T , by Proposition 2.3,
it follows that T(σT)s and T(σT)r are disjoint wherever |r| 6= |s|.
Question. Is there a Kronecker measure σ ∈ P(R+) such that I(Tσ) is
uncount-able?
This question is to be compared with Ryzhikov's question whether there is a weakly mixing, non-mixing ows with uncountable group of self-similarities, see [3], Prob-lem (1).
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Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toru«, Poland
E-mail address: fraczek@mat.umk.pl
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toru«, Poland
E-mail address: joanna.kulaga@gmail.com
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toru«, Poland
